Section 5-1: Special Segments in Triangles

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1 Section 5-1: Special Segments in Triangles Objectives: Identify medians, altitudes, angle bisectors, and perpendicular bisectors. perpendicular bisector C median altitude Vocabulary: A B Perpendicular bisector: a bisector of a segment that is perpendicular to the segment. Median: (in a triangle) a segment that joins a vertex of the triangle and the midpoint of the opposite side. Altitude: a segment from a vertex of the triangle of the line containing the opposite side perpendicular to the line containing that side.

2 Theorem: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segments. Angle bisector: a line segment that divides an angle into two congruent angles. Theorem: Any point on the bisector of an angle is equidistant from the sides of the angle.

3 Given points S(-4,3), G(-1, -1), and B(3, 2), find the median of GB, and determine the altitude from S to GB (hint: find the negative reciprocal of the slope of opposite sides). S G B midpoint formula: x 2 +x 1, y 2 +y 1 ( 2 2 ) distance formula: Answer: midpoint: (1.5, 2 ) distance: 5

4 Section 5-2: Right Triangles Objectives: (1)Use tests for congruence of right triangles. Theorem 5-5 (LL): If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. PROOF Given: ΔABC and ΔDEF are right triangles B and E are right angles AB DE BC EF Prove: ΔABC ΔDEF Statements Reasons (1)ΔABC and ΔDEF are right triangles (1) Given B and E are right angles AB DE BC EF (2) B E (2) All rt. angles are congruent. (3) ΔABC ΔDEF (3) SAS A B C D E F

5 Theorem 5-6 (HA): If the hypotenuse and an acute angle of one right triangle are congruent to hypotenuse and an acute angle of another right triangle, then the triangles are congruent. Theorem 5-7 (LA): If one leg and an acute angle of one right triangle are congruent to one leg and an acute angle of another right triangle, then the triangles are congruent. Postulate 5-1 (HL): If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

6 Find x and y so that ΔMNO ΔRST N 58 o O T M (47-8x)cm 15cm S (3y-20) o R Answers: By LA, ΔMNO ΔRST for x=4 and y=26

7 Geometry Section 5-3 Indirect Proof and Inequalities Indirect reasoning: Reasoning that assumes the conclusion false and then shows that this assumption leads to a contradiction of the hypothesis or some other accepted fact, like a postulate, theorem, or corollary. Then since the assumption has been proven false; the conclusion must be true. Indirect proof: Proof by contradiction. In an indirect proof one assumes that the statement to be proven is false. Then use logical reasoning to deduce a statement that contradicts a postulate or theorem. Once the contradiction is obtained, one concludes that the statement assumed false must in fact be true.

8 Steps to Indirect Proof: Step 1. Assume that the conclusion is false. Step 2. Show that the assumption leads to a contradiction of the hypothesis. Step 3. Point out that the assumption must be false and, therefore the conclusion must be true. State the Assumption a) The number 117 is divisible by 13 b) Hyrum is the best candidate in the election c) AB is not the median of ΔACD d) m c is less than m d

9 Exterior Angle Inequality Theorem: If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles o 56 o 54 o 70 o

10 Definition of Inequality: For any real numbers a and b, a > b if and only if there is a positive number c such that a=b+c. Properties of Inequalities for Real Numbers for All Numbers a, b, and c Comparison Prop: a < b, a=b or a > b Transitive Prop: if a b and b c then a c if a b and b c then a c Add. and Sub. Prop:if a b then a+c b+c and a-c b-c if a b than a+c b+c and a-c b-c Multi. and Divi. Prop: if c 0 and a b than ac bc and a/c b/c if c 0 and a b than ac bc and a/c b/c if c 0 and a b than ac bc and a/c b/c if c 0 and a b than ac bc and a/c b/c

11 Geometry Section 5.4 Inequalities for sides and angles of a triangle Objectives: Recognize and apply relationships between sides and angles in a triangle. Theorem 5 9: If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. B A BC > AC so A > B C Theorem 5 10: If an of a > another, the sides opposite the greater longer than the side opposite the longer. A is A > B so BC > AC B C

12 Theorem 5 11: The perpendicular segment from a point to a line is the shortest segment from the point to the line. (Corollary 5 1): the perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

13 Draw ΔRST with vertices R(-2,4), S(-5,-8), and T(6,10). List the angles in order from the greatest measure to least measure. RS= RT= ST=

14 Geometry Section 5 5 The Triangle Inequality Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the third side > >7 10+7>5 A. 18,45,21 B. 18,45,52 C. 18,21,52 D. 45,21,52 NO YES NO YES

15 The length of two sides of a triangle are 8 and 13. What are the possible length of the third side? 8+13> t 21>t 8+t>13 t>5 13+t>8 t> 5 5<t< You Try: The length of two sides of a triangle are 10 and 15. What are the possible length of the third side? 5<t<25

16 Given a rope 10 units in length, how many triangles can be made? (If you can only use whole numbers)

17 Geometry 5 6 Inequalities Involving Two Triangles objective: to apply SAS Inequality SSS Inequality SAS Inequality (Hinge Theorem) : (SAS TH) If two sides of one triangle are congruent to two sides of another triangle and the included angle in one triangle has a grater measure than the included angle in the other, then the third side of the first triangle is greater then the third in the second. A C D A D Then BC DF > AB DE B E F AC DF BC>EF

18 SSS Inequality Theorem: ( SSS = th. ) If two sides of one triangle are to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the point of sides in the first triangle is greater than the corresponding angle in the second triangle. D Given: AB DE AC DF BC > EF Then: A > D B A C E F

19 4 C F 3 D 3 How are they related? A. m ADC and m ADB m ADB > m ADC A 5 B B. m AFB and m BFD m AFB > m BFD

20 T U SW WZ TU TW UZ m UTW > m SWT S 57 o (3x + 9) W Z

Chapter 6 Summary 6.1. Using the Hypotenuse-Leg (HL) Congruence Theorem. Example

Chapter 6 Summary 6.1. Using the Hypotenuse-Leg (HL) Congruence Theorem. Example Chapter Summary Key Terms corresponding parts of congruent triangles are congruent (CPCTC) (.2) vertex angle of an isosceles triangle (.3) inverse (.4) contrapositive (.4) direct proof (.4) indirect proof